Fourier series convergence in $L^2(-\pi,\pi)$

Question

I've got a proof in my course of the fact that Fourier series converge in $L^2(\pi,\pi)$. But, since $L^2(-\pi,\pi)$ is a Hilbert space with the scalar product $\left<f,g\right>=\frac{1}{2\pi}\int_{-\pi}^\pi f\bar g$ and that $\{e^{inx}\}_{n\in\mathbb Z}$ is an orthonormal basis, don't we have that $S(f)=\sum_{n\in\mathbb Z}a_ne^{inx}$ where $a_n=\left<f,e^{inx}\right>$ converge in $L^2(-\pi,\pi)$ by definition ?